## Force of a set
by This is an abridged version of the paper submitted to the VII European Union Contest for Young Scientists in Newcastle, where it was awarded a third prize.
Many readers of Delta may have been amazed by the fact that a
segment and a square are equipotent sets, i.e., that there exists
a bijection between them (Fig. 1).
Fig.1 No wonder, since the discoverer of this fact himself wrote to Dedekind in a letter: "I see it, but I don't believe it". Both sets differ in dimension, but, on the other hand, dimension identifies e.g. an open segment with a closed one, whereas the latter has two more points than the first, doesn't it?
In this paper we introduce the notion of the force of a set. This
is to be a characteristic feature of some sets that takes into
account the intuitions suggested above and is in fact a combination
of dimension and Euler's characteristic. The force of a set will
always be a polynomial in one variable
Let's assume first that the force of the set of all real numbers
is
Fig. 2
Fig. 3
Let's make our considerations precise. A finite CW-complex is,
intuitively speaking, a space which is the union of a finite family
of disjoint, "well-behaved" cells (i.e. sets homeomorphic
with cube of different dimensions). In particular, a finite CW-complex
is compact and the border of each cell is the union of a subset
of cells of lower dimension. Figure 4 should give you an idea
of what it's all about. More pedantic readers may find the exact
definition in algebraic topology textbooks (e.g. in [2]).
Fig. 4
Define a function
The sum extends over all the cells of the complex
The definition is quite general; by appropriate choices of
Fig. 5 If the above considerations are not restricted to CW-complexes, we can similarly define other notions, like the measure of a set, probability or cardinality. Such definitions may not be very practical, but they show the generality of the introduced notion.
One might presume that in order to make the definitions of force
suggested by Figures 2 and 3 precise,
Fig. 6
Note that in the above examples all the forces of a given set
are of the same degree and, moreover, they are congruent modulo
take different values at The proof consists in the consideration of some simple cases and is therefore omitted.
We need not change the definition of the function
As can be seen, (see
also Fig. 6). But Euler's characteristic of
When two sets have identical CW-complex partitions, then their forces are clearly equal. The notion of force is so apt and natural, that the inverse theorem holds true as well.
Figure 7 shows how the theorem works. We invite our readers to provide a proof (which may be somewhat technical) by themselves. The following lemma may be a useful suggestion. Fig. 7
And one more thing: the set on the left of Fig. 7 is not a CW-complex. In fact, Euler's characteristic and the force of a set seem to work in some other cases too, e.g. for partitions which do not consist of CW-complexes (see Fig. 4) or for sets that are not compact. The authors ignore how the class of CW-complexes can be extended to cover such cases, too.
Fig. 8
[1] E. H. Spanier,
[2] A. T. Fomenko, D. B. Fuchs, V. L. Gutenmacher,
[3] K. Kuratowski,
[4] R. Duda, |